As a part of the 1992 -- 1993 Special Year on Combinatorial
Optimization at DIMACS,
László Lovász, Bruce Reed, and I organized a
workshop on perfect graphs. The workshop took place at Princeton
University on June 10--14, 1993. In February of that year, Bruce and
I prepared a list of open problems, which was then sent to all the invited
participants. The next version of the list, updated just before the
workshop, and still available at

are solvable in polynomial time for perfect graphs. Their algorithms
are based on the ellipsoid method and a polynomial time separation
algorithm for a certain class of positive semidefinite matrices
related to Lovász' upper bound (On the Shannon capacity of
a graph, IEEE Trans. Inform. Theory 25 (1979),
1--7; MR
81g:05095) on the Shannon capacity of a graph.

Can these four optimization problems be solved for perfect graphs
by polynomial-time algorithms of purely combinatorial nature avoiding
the numerical instability of the ellipsoid method? The decomposition
theorem for perfect graphs does not -- or at least does not yet --
provide such algorithms; warm-up exercises consist of trying to
design such algorithms for restricted classes of perfect graphs. One
of these infinitely many warm-up problems is singled out here.

A graph is called perfectly ordered if its vertex-set
is endowed with a linear order < in such a way that no
induced P4 with vertices a,b,c,d and edges
ab,bc,cd has a<b and d<c.
Chvátal (Perfectly ordered graphs. Topics on perfect
graphs, North-Holland Math. Stud., 88, North-Holland, Amsterdam-New
York, 1984, pp. 63--65. MR
86j:05059) has shown that, given a perfectly ordered
graph G and its coloring -- by some number k
of colors -- constructed by the familiar greedy algorithm, one can
find a clique of k vertices in G in
polynomial time; it follows that perfectly ordered graphs are
perfect.

Problem 2.1. Design a polynomial-time algorithm of purely
combinatorial nature that, given a perfectly ordered graph
G, constructs a largest stable set in G.

A small transversal in a graph G is a set
of alpha(G)+omega(G)-1 vertices which meets
all cliques of size omega(G) and all stable sets of size
alpha(G). The following problem is an easier variation on a
conjecture contributed to the 1993 workshop by Gurvich and Temkin and
on two conjectures proposed by Bacs&oacute, Boros, Gurvich, Maffray,
and Preissmann (On minimal imperfect graphs with circular
symmetry, J. Graph Theory 29 (1998),
209--225. MR
2000h:05116).

Conjecture 3.1. Every partitionable graph G with
alpha(G)>2 and omega(G)>2
has a small transversal or else contains a hole of length five.

One of the milestones in the development of our understanding of
perfect graphs was the theorem of Lovász (A characterization
of perfect graphs, J. Combinatorial Theory Ser. B 13
(1972), 95--98. MR
46 #8885) asserting that every minimal imperfect graph
G has precisely

alpha(G)omega(G) +1

vertices. This theorem implies that every minimal imperfect graph is
partitionable and that -- as pointed out by Chvátal (Notes on
perfect graphs, Progress in combinatorial optimization (Waterloo,
Ont., 1982), Academic Press, Toronto, Ont., 1984, pp. 107--115. MR
86h:05091) -- no minimal imperfect graph contains a small
transversal. It follows that a proof of Conjecture 3.1 would provide
another proof of the Strong Perfect Graph Theorem.

A partitionable graph without a small transversal has been constructed
by by Chvátal, Graham, Perold, and Whitesides (op.cit.). Its
vertices are 0,1,...,16; vertices i and
j are adjacent if and only if |i-j|mod 17 is one of

1,3,4,5,12,13,14,16.

Ara Markosian claims (here)
that this is the only known partitionable graph without a small
transversal. One of the many holes of length five in this graph is

1 - 4 - 8 - 12 - 15 - 1

Helpful hints in the direction of Conjecture 3.1 are the results of
Bacs&oacute, Boros, Gurvich, Maffray, and Preissmann (op.cit.); these
involve the construction of partitionable graphs with rotational
symmetries designed by Chvátal, Graham, Perold, and Whitesides
(op.cit.), which goes as follows.

Take any factorization

n-1=m1m2... m2k

into integer factors mj all greater than one;
define sets M1,M2,...
,M2k by

Mi={
tm1m2... mi-1: t=0,1,...
,mi-1}

and write

A=M1+M3+...M2k-1,
B=M2+M4+...M2k

The partitionable graph, with vertices v0 ,
v1,..., vn-1, has

alpha=m1m3...m2k-1 and
omega=m2m4...m2k ;

with subscript arithmetic modulo n, its stable sets of size
alpha are the n sets

{vj+a:a in A} with j=1,2, ... ,n

and its cliques of size omega are the n sets

{vj-b:b in B} with j=1,2, ... ,n.

This may not specify the graph completely: for instance, if
n=10 and m1=m2=3, then
A={0,1,2}, B={0,3,6}, and
so each vi may or may not be adjacent to
vi+5. More generally, following
Frédéric Maffray and Myriam Preissmann, a pair of vertices
is called indifferent if it is contained in no clique of
size omega and in no stable set of size
alpha; indifferent pairs may or may not be adjacent in the
CGPW graph.

Bacs&oacute, Boros, Gurvich, Maffray, and Preissmann
(op. cit.) say that a CGPW graph is of Type 2 if it has both
of the properties

m1=m3=...=m2k-1=2

and

m2=m4=...=m2k=2;

they say that the graph is of Type 1 if it has precisely one
of these properties; they say that the graph is of Type 0 if
it has neither of them. They prove that

every CGPW graph G of Type 0 or 1 with
alpha(G)>2 and omega(G)>2 has a small
transversal and

A helpful result on small transversals (called the "Parent Lemma"
in the paper of Bacs&oacute, Boros, Gurvich, Maffray, and
Preissmann) goes as follows. In a partitionable graph G, a
clique C of size omega(G) is a mother
of a vertex v if no clique of size omega(G)
meets C in the single vertex v; a stable set
S of size alpha(G) is a father of a
vertex v if no stable set of size alpha(G) meets
S in the single vertex v. Gurvich and Temkin
(Berge's conjecture holds for rotational graphs.(Russian)
Dokl. Akad. Nauk 332 (1993), 144--148. MR
94m:05156) and, independently, Bacs&oacute proved that

if some vertex in a partitionable graph has both parents,
then the graph has a small transversal.

The following problem, related to Conjecture 3.1, has been contributed to
the 1993 workshop by András Sebö.

Problem 3.2. True or false? If a partitionable graph
G contains two cliques of size omega(G) with
omega(G)-1 vertices in common and two stable sets of size
alpha(G) with alpha(G)-1 vertices in common, then
G has a small transversal.

if I(G)=I(G'), then G and G'
are either both perfect or both imperfect

in the hope that an elegant and easily verifiable certificate of perfection of
G could be expressed exclusively in terms of some such natural
invariant.

The finest invariant with properties (i) and (ii) assigns to each
graph G the unordered pair consisting of G
and its complement; the coarsest invariant with properties (i) and
(ii) sets

I(G)=1 if G is perfect and
I(G)=0 if G is imperfect.

One natural invariant that interpolates between these two extremes is the
P4-structure of G; this
invariant is defined as the hypergraph whose vertex-set V is
vertex-set of G and whose edges are the subsets of
V that induce P4's in
G. For example, here are two graphs with the same
P4-structure,

Additional graph invariants with properties (i) and (ii),
found by Chính Hoàng, are

the co-paw-structure of
G, defined as the hypergraph whose vertex-set
V is vertex-set of G and whose edges are the
subsets of V that induce C5 or
the paw (the graph with vertices a,b,c,d and edges
ab,ac,bc,cd) or the complement of the paw in
G (C. T. Hoàng, On the
disc-structure of perfect graphs. I. The co-paw-structure,
Proceedings of the Third International Conference on Graphs and
Optimization, GO-III (Leukerbad, 1998), Discrete Appl. Math.
94 (1999), 247--262. MR
2001a:05065);

the
co-C4-structure
of G, defined as the hypergraph whose vertex-set
V is vertex-set of G and whose edges are the
subsets of V that induce C5 or
C4 or the complement of
C4 in G
(C. T. Hoàng, On the disc-structure of perfect
graphs. II. The co-C4-structure, Discrete
Math. 252 (2002), 141--159);

the
co-P3-structure of
G, defined as the hypergraph whose vertex-set
V is vertex-set of G and whose edges are the
subsets of V that induce P3 or
the complement of P3 in G
(C. T. Hoàng, On the
co-P3-structure of perfect graphs, to appear in
SIAM J. Discrete Math.);

the
co-K3-structure of
G, defined as the hypergraph whose vertex-set
V is vertex-set of G and whose edges are the
subsets of V that induce K3 or
the complement of K3 in G
(since K3,P3,co-P3, and
co-K3 are the only graphs with three vertices,
two graphs have the same
co-P3-structure if and only if
they have the same
co-K3-structure).

where the subscripts are taken modulo k. It is easy, if a
little tedious, to show that a graph is a Berge graph if and only if
its P4-structure contains no induced odd ring;
this fact led in the past to speculations about possible certificates,
elegant, easily verifiable, and expressed exclusively in terms of
4-uniform hypergraphs, that a prescribed
P4-structure contains no induced odd ring.

Finding such a certificate is at least as difficult as the -- as
yet unsolved -- problem of showing that the class of Berge graphs
belongs to NP. A less ambitious program would be an attempt to
reformulate the decomposition
theorem for perfect graphs by Chudnovsky, Robertson, Seymour, and
Thomas exclusively in terms of
P4-structure.

The decomposition theorem asserts that

every Berge graph either belongs to one of five classes of basic Berge
graphs, namely,

bipartite graphs,

complements of bipartite graphs,

line-graphs of bipartite graphs,

complements of line-graphs of bipartite graphs,

"double split graphs",

or else it has one of four structural faults, namely,

2-join,

2-join in the complement,

M-join,

a balanced skew partition

(for definitions, see the paper by
Chudnovsky, Robertson, Seymour, and Thomas); in her thesis, Chudnovsky
showed that the stronger assertion with M-joins dropped remains valid.
Do the remaining three structural faults admit natural counterparts
formulated exclusively in terms of
P4-structure?

The following two problems represent one attempt to state this
question in precise terms.

Problem 4.1. Find a class C1 of
4-uniform hypergraphs such that

the P4-structure of a graph
belongs to C1 if and only if
it is the P4-structure of a (possibly
different) graph
such that this graph or its complement admits a 2-join

membership in C1 can be tested in
polynomial time.

Problem 4.2. Find a class C2 of
4-uniform hypergraphs such that

the P4-structure of a graph
belongs to C2 if and only if
it is the P4-structure of a (possibly
different) graph
that admits a balanced skew partition,

membership in C2 can be tested in
polynomial time.

It is conceivable that these problems can be solved by building up on
the following three results:

and he defined a perfectly contractile graph as
any graph G such that all induced subgraphs of
G are even-contractile. (The results of Meyniel
guarantee that every perfectly contractile graph is perfect.)

Hazel Everett and Bruce Reed contributed to the 1993 workshop a
number of conjectures on even pairs and perfectly contractile graphs;
two of them are singled out here. The odd stretcher in the
first of them is any graph that consists of three vertex-disjoint
triangles and three vertex-disjoint paths, each path having an odd
number of edges and one endpoint in each of the two triangles. (In
particular, the smallest odd stretcher is the antihole with six
vertices.)

Conjecture 5.1. A graph is perfectly contractile if and only if
it contains no odd hole, no antihole, and no odd stretcher.

Conjecture 5.2. Every perfectly contractile graph other than a
clique contains an even pair whose contraction leaves the graph
perfectly contractile.

Bienstock (On the complexity of testing for odd holes and
induced odd paths, Discrete Math. 90 (1991),
85--92. MR
92m:68040a Corrigendum: Discrete Math. 102
(1992), 109. MR
92m:68040b) has proved that it is coNP-complete to
determine if a graph contains an even pair. By analogy with the notion
of en even pair, an odd pair is a pair of nonadjacent
vertices such that every chordless path between them has an odd number
of edges. The following two algorithmic problems are obvious.

Problem 5.3. Design a polynomial-time algorithm to determine
if a perfect graph contains an even pair.

Problem 5.4. Design a polynomial-time algorithm to determine
if a perfect graph contains an odd pair.

Other problems on path parity and perfection, including additional
conjectures contributed to the 1993 workshop by Everett and Reed, can
be found in

Problem 6.1. Prove the existence of a function g such
that
every graph G with no odd hole has chromatic number at most
g(omega(G)).

Guoli Ding asked whether g(3)=4. His question has been
answered in the affirmative by Robertson, Seymour, and Thomas, who
provided a structural characterization of graphs without odd holes and
without K4.

Problem 6.2. For all positive integers k, prove the
existence of a function gk such that every graph
G with no odd hole longer than k has chromatic
number at most gk(omega(G)).

Problem 6.3. For all positive integers k, prove the
existence of a function hk such that every graph
G with no hole longer than k has chromatic
number at most hk(omega(G)).

Chính Hoàng and
Colin McDiarmid (On the divisibility of graphs, Discrete
Math. 242 (2002), 145--156. MR 1
874 761) say that a graph is 2-divisible
if, for each of its induced subgraphs F, the vertex-set
of F can be partitioned into subsets
S1 and S2 in such a
way that every largest clique in F meets both
S1 and S2. Trivially,
no odd hole is 2-divisible, and so every 2-divisible graph is
odd-hole-free; Hoàng and McDiarmid conjectured the converse:

Conjecture 6.4. Every odd-hole-free graph is 2-divisible.

Hoàng and McDiarmid proved Conjecture 6.4. for claw-free graphs
and noted that it also holds for graphs without K4 (since
g(3)=4 in Problem 6.1).

Can something similar be said about graphs without even holes? The
notion of 2-divisibility generalizes: Hoàng and McDiarmid say
that a graph is k-divisible if, for each of its induced
subgraphs F, the vertex-set of F can be
partitioned into subsets S1,
S2, ..., Sk in such a
way that no Si contains a largest clique in
F.

Conjecture 6.5. (Hoàng) Every even-hole-free
graph is 3-divisible.

Conjecture 6.6. (Hoàng) Every even-hole-free
graph G is
colorable by

2·omega(G)-1

colors.

Conjecture 6.6 implies Conjecture 6.5: more generally, if
G is a graph and
k is an integer such that

Problem 7.1. For every (undirected) graph G, let
H1(G) denote the graph whose vertices are the
triangles in G, two vertices of H1(G)
being adjacent if and only if the corresponding triangles in
G share an edge. For which graphs G is
H1(G) perfect?

Problem 7.2. For every directed graph G, let
H2(G) denote the graph whose vertices are the
transitive triangles in G, two vertices of
H2(G) being adjacent if and only if the
corresponding triangles in G share an arc. For which
graphs G is H2(G) perfect?

Problem 7.3. For every directed graph G, let
H3(G) denote the graph whose vertices are the
cyclic triangles in G, two vertices of
H3(G) being adjacent if and only if the
corresponding triangles in G share an arc. For which
graphs G is H3(G) perfect?

If G is a planar oriented graph, then
H3(G) is known to be
(K4-e)-free Berge graph, and hence perfect.

Comment by Leizhen Cai: for each pair of graphs X and
Y, Cai, Corneil and Proskurowski (A generalization of
line graphs: (X,Y)-intersection graphs, J. Graph
Theory 21 (1996), no. 3, 267--287. MR
96m:05164) defined the (X,Y)-intersection
graph of a graph G as the graph whose vertices
correspond to distinct induced subgraphs of G that are
isomorphic to Y, and where two vertices are adjacent iff
the intersection of the corresponding subgraphs contains an induced
subgraph isomorphic to X. In this terminology,
H1(G) is the
(K2,K3)-intersection graphs of
G. Cai, Corneil and Proskurowski have shown that the
family of (X,Y)-intersection graphs is a subfamily of
line-graphs of k-uniform hypergraphs, where k is
the number of distinct induced copies of X in Y.